Question: The sum of two numbers is $31$, and their difference is $11$. What are the two numbers?
Answer: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 31}$ ${x-y = 11}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 42 $ $ x = \dfrac{42}{2} $ ${x = 21}$ Now that you know ${x = 21}$ , plug it back into $ {x+y = 31}$ to find $y$ ${(21)}{ + y = 31}$ ${y = 10}$ You can also plug ${x = 21}$ into $ {x-y = 11}$ and get the same answer for $y$ ${(21)}{ - y = 11}$ ${y = 10}$ Therefore, the larger number is $21$, and the smaller number is $10$.